On the condition number of the Vandermonde matrix of thenth cyclotomic polynomial

Abstract

Recently, Blanco-Chacón proved the equivalence between the Ring Learning With Errors and Polynomial Learning With Errors problems for some families of cyclotomic number fields by giving some upper bounds for the condition number Cond(Vn) of the Vandermonde matrix Vn associated to the nth cyclotomic polynomial. We prove some results on the singular values of Vn and, in particular, we determine Cond(Vn) for n = 2kpℓ, where k, ℓ ≥ 0 are integers and p is an odd prime number.

RLWE/PLWE equivalence for the maximal totally real subextension of the $2^rpq$-th cyclotomic field

“…As justified in [16] and later in [2], [3] and [17], the idea of noise increase can be formally captured by means of the condition number, defined in terms of the Frobenius norm:…”

“…In the cyclotomic scenario, which is the most dealt with in practical cryptographic primitives, the problem is now reasonably well understood (cf. [6], [2], [17]).…”

Trace-based cryptanalysis of cyclotomic $R_{q,0}\times R_q$-PLWE for the non-split case

“…However, PLWE is more suitable for efficient implementations thanks to very fast multiplication algorithms like Toom, Karatsuba or versions of the Number Theoretic Transform (NTT) which are not available for number fields, where just finding integral bases becomes cumbersome even for moderately large degree and discriminant (let alone those of cryptographic size). Luckily, in a good number of interesting cases both problems are equivalent (see [2,3,5,6,13,14,15]).…”

RLWE and PLWE over cyclotomic fields are not equivalent

“…Blanco-Chacón [6] (see also [8,9]) proved that Cond(V n ) = O(n r k ), where r k > 0 is a constant depending only on the number k of distinct prime factors of n. Therefore, RLWE and PLWE restricted to the positive integers n with a bounded number of prime factors are equivalent. Furthermore, in a previous work [12], the authors gave an explicit formula for the condition number of V n when n is a prime power or a power of 2 times an odd prime power. Our main result is the following.…”